Selected Problems from the Final Round of National Olympiad

Question

Juku discovered that of the things in his satchel,

60%

were ugly and

76%

were useless. He scrapped all things that were both ugly and useless, and added things that were both beautiful and useful. After this, of the things in Juku's satchel,

25%

are ugly and

45%

are useless. How many percent of the things in Juku's satchel were both beautiful and useful initially?

Answer:

4%

.

Accepted Answer

Observe that the amount of things that are beautiful but useless and things that are useful but ugly remained unchanged. The difference between the percentages of these things was 76% - 60% = 16% before displacement but is 45% - 25% = 20% after that. Hence the overall number of things in the satchel decreased 20 : 16 = 1.25 times. Things that are beautiful but useless form 45% of all things after the displacement, hence they formed 45% : 1.25 = 36% before it. As there were 100% - 60% = 40% of beautiful things in total, the things that were both beautiful and useful constituted 40% - 36% = 4% of the content of the satchel.